*Modular Arithmetic*: This particular cardigan has an 18-row cable pattern that repeats several times beginning with row 9 of the sweater body, while at the same time you knit a buttonhole every tenth row beginning in row 5. So I know that I need to put a buttonhole in row 5, 15, 25, 35, and 45. I want to start counting my rows with row 1 of the CABLE PATTERN, so using the new numbering system for rows, I'll be putting buttonholes in row 7 (this is the second buttonhole), 17, 27, and 37. Then using modular arithmetic (also known as clock arithmetic) I reduce those modulo 18 and work buttonholes in rows 7, 17, 9, and 1 of the CABLE PATTERN. For me, this is much easier than trying to keep track of one count for the cable pattern and another for the buttonholes - instead, I just track everything in terms of the cable pattern.

*Symmetry*: If you look closely at my photos in this post and compare them to the photos in the pattern, you'll notice that I changed some of the cable crossings. Many mathematicians care a lot about symmetry (non-mathematicians care about this too, of course, but we're trained to notice it wherever we can). This is a case where I think the pattern-writer was wrong. If you imagine a vertical line going down the middle of the back of the sweater, in the center seed stitch column, and think of reflecting one side of the sweater across that line, you would get the other side of the sweater. This is called a reflectional symmetry, and it makes for a much more pleasing image than what is written in the original pattern, with all of the large fancy cables twisting to the "right" and all of the little cables twisting "left." I fixed this so that the two large fancy cables on each of the front and back twist toward the center, and each of the little cables twists toward the large fancy cable it frames.

*Braids*: This one doesn't really improve my knitting as much as add to my enjoyment. My research is in knot theory, which is closely related to the study of mathematical braids. Every knitted cable is a braid; in this sweater, each of the fancy cables is a two-strand braid, and each of the little cables is a four-strand braid. Referring back to symmetry for a moment, the mirror image of a braid is its inverse, so in this sweater we see braids paired with their inverses. It makes me so happy when my work shows up in other areas of my life! I'm so glad I'm a mathematician - if I wasn't, I wouldn't be able to properly appreciate this little sweater!

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